Finite Group Databasebeta

← #33129916

Subgroups — #33129916

Order 18688.

940 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988742C211yesmaximal, characteristic
2C2×C4×#11924018C211yesmaximal, characteristic, complemented
3C22⋊C4×Dic146C211yesmaximal, characteristic
4#22611470C211yesmaximal, characteristic, complemented
5#22611362C211yesmaximal, characteristic
6#22611253C211yesmaximal, characteristic, complemented
7#22611131C211yesmaximal, characteristic
8#22610606C211yesmaximal, characteristic, complemented
9#22610539C211yesmaximal, characteristic
10#22610275C211yesmaximal, characteristic, complemented
11#22610272C211yesmaximal, characteristic, complemented
12#22610184C211yesmaximal, characteristic, complemented
13#22609933C211yesmaximal, characteristic, complemented
14#22609865C211yesmaximal, characteristic
15C73×#2423C211yesmaximal, characteristic
16C23×Dic146C2211yescharacteristic
17C22×#11924018C2211yescharacteristic
18C22×#11924018C2211yescharacteristic
19C2×C4×Dic146C2211yescharacteristic
20C2×C4×Dic146C2211yescharacteristic
21#17988742—24no—
22C4×#11924018—24no—
23C2×#14556524C2211yescharacteristic
24C2×#14556524C2211yescharacteristic
25C2×C4⋊#10596831C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556514C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556514C2211yescharacteristic
31C2×#14556514C2211yescharacteristic
32C2×#14556514C2211yescharacteristic
33C2×#14556514C2211yescharacteristic
34C2×C22⋊#10596831C2211yescharacteristic
35C2×C22⋊#10596831C2211yescharacteristic
36C2×C4×#10596831C2211yescharacteristic
37#17988493C2211yescharacteristic
38#17988493C2211yescharacteristic
39#17988491—21no—
40#17988491—21no—
41#17988447C2211yescharacteristic
42#17988445—21no—
43C22⋊C4×Dic73—21no—
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988342C2211yescharacteristic
49#17988342C2211yescharacteristic
50C2×C22⋊Q8×C73C2211yescharacteristic
51C2×C4×Q8×C73C2211yescharacteristic
52#399×C73—21no—
53#398×C73—21no—
54#391×C73C2211yescharacteristic
55#389×C73C2211yescharacteristic
56#388×C73—21no—
57#387×C73—21no—
58#384×C73C2211yescharacteristic
59#380×C73C2211yescharacteristic
60C4×C22⋊C4×C73C2211yescharacteristic
61C22×Dic146—22no—
62C22×Dic146—22no—
63C22×Dic146—22no—
64C22×Dic146—22no—
65C22×Dic146—21no—
66C22×Dic146—41no—
67C22×Dic146—41no—
68C22×Dic146C2311yescharacteristic
69C22×Dic146C2311yescharacteristic
70C23×Dic73—21no—
71C2×#11924018—22no—
72C2×#11924018—22no—
73C2×#11924018—22no—
74C2×#11924018—22no—
75C2×#11924018—41no—
76C2×#11924018D414yes—
77C22×#10596831C2311yescharacteristic
78C22×#10596831C2311yescharacteristic
79C22×#10596831C2311yescharacteristic
80C22×#10596831C2311yescharacteristic
81C4×Dic146—22no—
82C4×Dic146—22no—
83C4×Dic146—41no—
84C4×Dic146—41no—
85C2×C4⋊Dic73C2311yescharacteristic
86C2×C4⋊Dic73C2311yescharacteristic
87C2×#11924007—21no—
88C2×#11924007—21no—
89C2×C22⋊Dic73—21no—
90C2×C4×Dic73—21no—
91C2×C4×Dic73—21no—
92#14556524—22no—
93#14556524—41no—
94C4⋊#10596831—41no—
95#14556515—22no—
96#14556515—41no—
97#14556514—22no—
98#14556514—22no—
99#14556514—22no—
100#14556514—22no—
101#14556514—22no—
102#14556514—41no—
103C22⋊#10596831—22no—
104C22⋊#10596831—41no—
105C4×#10596831—22no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498—21no—
111C22×Q8×C73C2311yescharacteristic
112C23×C4×C73C2311yescharacteristic
113C22⋊Q8×C73—22no—
114C22⋊Q8×C73—41no—
115C4×Q8×C73—22no—
116C4×Q8×C73—41no—
117C2×C4⋊C4×C73—21no—
118C2×C4⋊C4×C73—21no—
119C2×C4⋊C4×C73—21no—
120C2×C4⋊C4×C73—21no—
121C2×C4⋊C4×C73—21no—
122C2×C4⋊C4×C73C2311yescharacteristic
123C2×C4⋊C4×C73C2311yescharacteristic
124C2×C4⋊C4×C73C2311yescharacteristic
125C2×C22⋊C4×C73—21no—
126C2×C22⋊C4×C73—21no—
127C2×C22⋊C4×C73D411yescharacteristic
128C2×C22⋊C4×C73D411yescharacteristic
129C2×C42×C73—21no—
130C2×C42×C73C2311yescharacteristic
131C2×C42×C73C2311yescharacteristic
132#95×C73—21no—
133#95×C73—21no—
134#95×C73—21no—
135#95×C73—21no—
136#95×C73D411yescharacteristic
137#95×C73D411yescharacteristic
138C2×Dic146—21no—
139C2×Dic146—21no—
140C2×Dic146—21no—
141C2×Dic146—21no—
142C2×Dic146—21no—
143C2×Dic146—21no—
144C2×Dic146—21no—
145C2×Dic146—24no—
146C2×Dic146—21no—
147C2×Dic146—44no—
148C2×Dic146—44no—
149C2×Dic146—44no—
150C2×Dic146—42no—
151C2×Dic146—42no—
152C2×Dic146—44no—
153C2×Dic146C2×D412yes—
154C2×Dic146#4112yes—
155C2×Dic146#4112yes—
156C2×Dic146#4112yes—
157C22×Dic73—21no—
158C22×Dic73—24no—
159C22×Dic73—21no—
160C22×Dic73—21no—
161C22×Dic73—21no—
162C22×Dic73—42no—
163C22×Dic73—42no—
164#11924018—28no—
165#11924018—44no—
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831—21no—
169C2×#10596831—21no—
170C2×#10596831—21no—
171C2×#10596831—21no—
172C2×#10596831—21no—
173C2×#10596831—21no—
174C2×#10596831—21no—
175C2×#10596831—21no—
176C2×#10596831—21no—
177C2×#10596831C2×D412yes—
178C4⋊Dic73—21no—
179C4⋊Dic73—21no—
180C4⋊Dic73—21no—
181C4⋊Dic73—21no—
182#11924007—42no—
183#11924007—42no—
184C22⋊Dic73—42no—
185C4×Dic73—42no—
186C4×Dic73—42no—
187C24×C73C2×D411yescharacteristic
188C2×Q8×C73—21no—
189C2×Q8×C73—21no—
190C2×Q8×C73—21no—
191C2×Q8×C73—21no—
192C2×Q8×C73—21no—
193C2×Q8×C73C2×D412yes—
194C22×C4×C73—21no—
195C22×C4×C73—21no—
196C22×C4×C73—21no—
197C22×C4×C73—21no—
198C22×C4×C73—21no—
199C22×C4×C73—21no—
200C22×C4×C73—21no—
201C22×C4×C73—21no—
202C22×C4×C73—21no—
203C22×C4×C73—21no—
204C22×C4×C73C2×D411yescharacteristic
205C22×C4×C73C2×D411yescharacteristic
206C22×C4×C73C2×D411yescharacteristic
207C22×C4×C73C2×D411yescharacteristic
208C22×C4×C73C2×D411yescharacteristic
209C22×C4×C73C2411yescharacteristic
210C4⋊C4×C73—21no—
211C4⋊C4×C73—21no—
212C4⋊C4×C73—21no—
213C4⋊C4×C73—21no—
214C4⋊C4×C73—21no—
215C4⋊C4×C73—21no—
216C4⋊C4×C73—42no—
217C4⋊C4×C73—42no—
218C4⋊C4×C73—42no—
219C4⋊C4×C73—81no—
220C4⋊C4×C73—81no—
221C22⋊C4×C73—41no—
222C22⋊C4×C73—42no—
223C22⋊C4×C73—81no—
224C22⋊C4×C73Q1614yescomplemented
225C42×C73—21no—
226C42×C73—21no—
227C42×C73—21no—
228C42×C73—21no—
229C42×C73—42no—
230C23×C73—21no—
231C23×C73—21no—
232C23×C73—21no—
233C23×C73—21no—
234C23×C73—21no—
235C23×C73—21no—
236C23×C73C2×Q1612yes—
237C23×C73C22×D411yescharacteristic
238Q8×C73—22no—
239Q8×C73—22no—
240Q8×C73—41no—
241Q8×C73—41no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—22no—
249C2×C4×C73—21no—
250C2×C4×C73—21no—
251C2×C4×C73—21no—
252C2×C4×C73—21no—
253C2×C4×C73—21no—
254C2×C4×C73—21no—
255C2×C4×C73—21no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73—41no—
259C2×C4×C73—41no—
260C2×C4×C73—41no—
261C2×C4×C73—41no—
262C2×C4×C73—41no—
263C2×C4×C73—41no—
264C2×C4×C73—41no—
265C2×C4×C73—41no—
266C2×C4×C73—41no—
267C2×C4×C73—41no—
268C2×C4×C73—41no—
269C2×C4×C73—41no—
270C2×C4×C73—41no—
271C2×C4×C73C2×Q1612yes—
272C2×C4×C73C2×Q1612yes—
273C2×C4×C73C22×D411yescharacteristic
274C2×C4×C73C2×#4111yescharacteristic
275C2×C4×C73C2×#4111yescharacteristic
276C2×C4×C73C2×#4111yescharacteristic
277C2×C4×C73H(5,𝔽2)11yescharacteristic
278C2×C4×C73H(5,𝔽2)11yescharacteristic
279C2×Dic73—22no—
280C2×Dic73—22no—
281C2×Dic73—22no—
282C2×Dic73—22no—
283C2×Dic73—48no—
284C2×Dic73—41no—
285C2×Dic73—41no—
286C2×Dic73—41no—
287C2×Dic73—41no—
288C2×Dic73—41no—
289C2×Dic73—41no—
290C2×Dic73—41no—
291C2×Dic73—41no—
292Dic146—22no—
293Dic146—22no—
294Dic146—22no—
295Dic146—22no—
296Dic146—22no—
297Dic146—48no—
298Dic146—41no—
299Dic146—41no—
300Dic146#12314yes—
301#10596831—22no—
302#10596831—41no—
303#10596831—41no—
304#10596831—41no—
305C22×C73—22no—
306C22×C73—22no—
307C22×C73—22no—
308C22×C73—41no—
309C22×C73—41no—
310C22×C73—41no—
311C22×C73—41no—
312C22×C73#52511yescharacteristic
313C22×C73D4211yescharacteristic
314C22×C73#54611yescharacteristic
315C22×C73C22×Q1611yescharacteristic
316C22×C73#57511yescharacteristic
317C22×C73#57711yescharacteristic
318C22×C73#57811yescharacteristic
319C4×C73—22no—
320C4×C73—22no—
321C4×C73—41no—
322C4×C73—42no—
323C4×C73—41no—
324C4×C73—41no—
325C4×C73—81no—
326C4×C73—81no—
327C4×C73—81no—
328C4×C73C2×#12311yescharacteristic
329C4×C73#53811yescharacteristic
330C4×C73#54611yescharacteristic
331C4×C73#55111yescharacteristic
332Dic73—24no—
333Dic73—42no—
334Dic73—81no—
335Dic73—81no—
336#24906—731nomaximal, Sylow
337C2×C73—42no—
338C2×C73#241811yescharacteristic
339C2×C73#310411yescharacteristic
340C2×C73#312111yescharacteristic
341C2×C73#327111yescharacteristic
342C2×C73D4×Q1611yescharacteristic
343C2×C73#329911yescharacteristic
344C2×C73#332711yescharacteristic
345C2×#451—731no—
346C2×C4×Q16—731no—
347#2423D7311yescharacteristic
348Q8×C22⋊C4—731no—
349#2095—731no—
350#2076—731no—
351#2056—731no—
352#2036—731no—
353#1933—731no—
354#1923—731no—
355#1878—731no—
356#1878—731no—
357#1863—731no—
358#1803—731no—
359#1787—731no—
360C73#2490611yescharacteristic, complemented, Sylow
361C23×Q8—731no—
362C22×Q16—731no—
363C22×Q16—731no—
364C2×C22⋊Q8C2×D7311yescharacteristic
365C2×C4×Q8—731no—
366C2×C4×Q8—731no—
367C2×C4×Q8C2×D7311yescharacteristic
368#451—1464no—
369C4×Q16—1464no—
370C2×#107—731no—
371C2×#107—731no—
372C2×C4⋊C8—731no—
373C2×#103—731no—
374C2×#103—731no—
375C2×#103—731no—
376C2×#103—731no—
377C2×#103—731no—
378C2×#103—731no—
379C2×#103—731no—
380C2×#103—731no—
381C2×C22⋊C8—731no—
382C2×C22⋊C8—731no—
383C2×C4×C8—731no—
384#399—21no—
385#398—21no—
386#391—731no—
387#391—731no—
388#391—1461no—
389#391—1461no—
390#391C2×D7311yescharacteristic
391#389C2×D7311yescharacteristic
392#388—21no—
393#387—21no—
394#384C2×D7311yescharacteristic
395#380—731no—
396#380—1461no—
397#380C2×D7311yescharacteristic
398C4×C22⋊C4—1461no—
399C4×C22⋊C4C2×D7311yescharacteristic
400#340—731no—
401#340—731no—
402#340—731no—
403#340—731no—
404#336—731no—
405#336—731no—
406C22×Q8—2921no—
407C22×Q8—2921no—
408C22×Q8—731no—
409C22×Q8—731no—
410C22×Q8—1462no—
411C22×Q8—1462no—
412C22×Q8—1462no—
413C22×Q8—1462no—
414C22×Q8—1461no—
415C22×Q8C22×D7311yescharacteristic
416C23×C4—1461no—
417C23×C4C22×D7311yescharacteristic
418C2×Q16—2921no—
419C2×Q16—734no—
420C2×Q16—1462no—
421C2×Q16—1462no—
422C2×Q16—1462no—
423C2×Q16—1462no—
424C22×C8—731no—
425C22×C8—731no—
426C22×C8—731no—
427C22×C8—731no—
428C22⋊Q8—22no—
429C22⋊Q8—41no—
430C4×Q8—22no—
431C4×Q8—41no—
432C4×Q8—2921no—
433C4×Q8—2921no—
434C4×Q8—1462no—
435C4×Q8—1462no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—731no—
442C2×C4⋊C4—731no—
443C2×C4⋊C4—1461no—
444C2×C4⋊C4—1461no—
445C2×C4⋊C4C22×D7311yescharacteristic
446C2×C4⋊C4C22×D7311yescharacteristic
447C2×C4⋊C4C22×D7311yescharacteristic
448C2×C22⋊C4—21no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4—1461no—
451C2×C22⋊C4C22⋊D7311yescharacteristic
452C2×C22⋊C4C22⋊D7311yescharacteristic
453C2×C42—21no—
454C2×C42—1461no—
455C2×C42—1461no—
456C2×C42C22×D7311yescharacteristic
457C2×C42C22×D7311yescharacteristic
458#107—2921no—
459#107—1462no—
460C4⋊C8—2921no—
461#103—2921no—
462#103—2921no—
463#103—1462no—
464#103—1462no—
465#103—1462no—
466#103—1462no—
467#103—1462no—
468#103—1462no—
469C22⋊C8—2921no—
470C22⋊C8—1462no—
471C4×C8—1462no—
472#95—21no—
473#95—21no—
474#95—21no—
475#95—21no—
476#95—1461no—
477#95—1461no—
478#95—1461no—
479#95—1461no—
480#95—1461no—
481#95C22⋊D7311yescharacteristic
482#95C22⋊D7311yescharacteristic
483C24C2×C22⋊D7311yescharacteristic
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—2922no—
490C2×Q8—2922no—
491C2×Q8—2924no—
492C2×Q8—2924no—
493C2×Q8—2924no—
494C2×Q8—2924no—
495C2×Q8—732no—
496C2×Q8—732no—
497C2×Q8—732no—
498C2×Q8—732no—
499C2×Q8—1464no—
500C2×Q8—1461no—
501C2×Q8—1461no—
502C2×Q8—1461no—
503C2×Q8—1461no—
504C2×Q8—1461no—
505C2×Q8—1461no—
506C2×Q8—1461no—
507C2×Q8—1461no—
508C2×Q8D4×D7312yes—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—2922no—
520C22×C4—2922no—
521C22×C4—1461no—
522C22×C4—1461no—
523C22×C4—1464no—
524C22×C4—1461no—
525C22×C4—1461no—
526C22×C4C2×C22⋊D7311yescharacteristic
527C22×C4C2×C22⋊D7311yescharacteristic
528C22×C4C2×C22⋊D7311yescharacteristic
529C22×C4C2×C22⋊D7311yescharacteristic
530C22×C4C2×C22⋊D7311yescharacteristic
531C22×C4C23×D7311yescharacteristic
532Q16—2924no—
533Q16—1468no—
534C2×C8—732no—
535C2×C8—1461no—
536C2×C8—1461no—
537C2×C8—1461no—
538C2×C8—1461no—
539C2×C8—1461no—
540C2×C8—1461no—
541C2×C8—1461no—
542C2×C8—1461no—
543C2×C8—1461no—
544C2×C8—1461no—
545C2×C8—1461no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—81no—
556C4⋊C4—81no—
557C4⋊C4—2922no—
558C4⋊C4—2922no—
559C4⋊C4—1461no—
560C4⋊C4—1461no—
561C4⋊C4—1461no—
562C4⋊C4—1461no—
563C22⋊C4—41no—
564C22⋊C4—42no—
565C22⋊C4—81no—
566C22⋊C4—2922no—
567C22⋊C4#1192401814yescomplemented
568C42—21no—
569C42—21no—
570C42—21no—
571C42—21no—
572C42—42no—
573C42—2922no—
574C42—2922no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23C2×#1192401812yes—
582C23C22×C22⋊D7311yescharacteristic
583Q8—22no—
584Q8—22no—
585Q8—41no—
586Q8—41no—
587Q8—2928no—
588Q8—2921no—
589Q8—2921no—
590Q8—734no—
591Q8—1462no—
592Q8—1462no—
593Q8—1462no—
594Q8—1462no—
595Q8—1462no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—2921no—
626C2×C4—2921no—
627C2×C4—2928no—
628C2×C4—2921no—
629C2×C4—2921no—
630C2×C4—2921no—
631C2×C4—2921no—
632C2×C4—2921no—
633C2×C4—2921no—
634C2×C4—1462no—
635C2×C4—1462no—
636C2×C4—1462no—
637C2×C4—1462no—
638C2×C4C2×#1192401812yes—
639C2×C4C2×#1192401812yes—
640C2×C4C2×D4×D7311yescharacteristic
641C2×C4#41×D7311yescharacteristic
642C2×C4#41×D7311yescharacteristic
643C2×C4#41×D7311yescharacteristic
644C2×C4#1455663611yescharacteristic
645C2×C4#1455663711yescharacteristic
646C8—2921no—
647C8—2921no—
648C8—2921no—
649C8—1462no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22#1798909411yescharacteristic
658C22D4×C22⋊D7311yescharacteristic
659C22#1798922511yescharacteristic
660C22C22×#1192401811yescharacteristic
661C22#1798938511yescharacteristic
662C22#1798939811yescharacteristic
663C22#1798940211yescharacteristic
664C4—22no—
665C4—22no—
666C4—41no—
667C4—41no—
668C4—42no—
669C4—41no—
670C4—81no—
671C4—81no—
672C4—81no—
673C4—2922no—
674C4—5841no—
675C4—5841no—
676C4—1464no—
677C4#123×D7311yescharacteristic
678C4#1798917311yescharacteristic
679C4#1798922911yescharacteristic
680C4#1798926511yescharacteristic
681C2—42no—
682C2#2261339011yescharacteristic
683C2#2261996611yescharacteristic
684C2#2262015411yescharacteristic
685C2#2262151211yescharacteristic
686C2D4×#1192401811yescharacteristic
687C2#2262180111yescharacteristic
688C2#2262219311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2423C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#2423D73
hypocenterC73#24906
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2423

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C4×#11924018C2semidirect
4. #22611470C2semidirect
6. #22611253C2semidirect
8. #22610606C2semidirect
10. #22610275C2semidirect
11. #22610272C2semidirect
12. #22610184C2semidirect
13. #22609933C2semidirect
224. C22⋊C4×C73Q16semidirect
360. C73#24906semidirect
567. C22⋊C4#11924018semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179887422
C2×C4×#119240182
C22⋊C4×Dic1462
#226114702
#226113622
#226112532
#226111312
#226106062
#226105392
#226102752
#226102722
#226101842
#226099332
#226098652
C73×#24232
#2490673